HAL Id: hal-01938818https://hal.archives-ouvertes.fr/hal-01938818

Submitted on 28 Nov 2018

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Mathematical analysis of informal arguments: Acase-study in teacher-training context

Renaud Chorlay

To cite this version:Renaud Chorlay. Mathematical analysis of informal arguments: A case-study in teacher-trainingcontext. CERME 10, Feb 2017, Dublin, Ireland. �hal-01938818�

Mathematical analysis of informal arguments: A case-study in

teacher-training context

Renaud Chorlay

LDAR, Uni. Paris Diderot, France & ESPE de l’Académie de Paris, France

renaud.chorlay@espe-paris.fr

We will endeavour to show that some historical documents which are probably too difficult to be used

in the classroom can nevertheless be fruitfully used in teacher-training, in order to provide teachers

with tools for the analysis of informal or semi-formal justifications. We will analyze an extract from

the Nine Chapters, so as to spell out tool for the analysis of three school documents bearing on the

multiplication of decimal or general fractions.

Keywords: History of mathematics, fractions, teacher-training, argumentation, generic examples.

Introduction

We will endeavour to show that some historical documents which are probably too difficult to be

used in the classroom can nevertheless be fruitfully used in teacher-training, in order to provide

teachers with tools for the analysis of informal or semi-formal justifications. We will analyze an

extract from the Nine Chapters, so as to spell out tool for the analysis of three school documents

bearing on the multiplication of decimal or general fractions.

A historical example: Multiplying fractions in the Nine Chapters

The Nine Chapters on the Mathematical Art (九章算術) is a Classic that was compiled during the

Han Dynasty (206 BC – 220 AD). It contains an organized list of problems with general procedures

to solve them, yet with no attempts at justification. In 263 AD, scholar Liu Hui wrote an extensive

commentary in which endeavoured to justify all the procedures, and identify key subprocedures of

general scope. We will comment on the passage in chapter one which deals with the multiplication

of fractions, on the basis of the critical edition and French translation (Chemla & Shuchun, 2004)1.

Problem 1.19. Now given a [rectangular] field, 4/7 bu in breadth and 3/5 bu in length. Tell: what is

the area? Answer: 12/35 bu.

(…)2 Procedure for the multiplication of parts: multiply the denominators to make up the divisor;

multiply the numerators to make up the dividend. Divide the dividend by the divisor.

(…) [Liu Hui]: In all cases when a dividend does not fill a divisor, then they are called denominator

and numerator. If there are parts, expand the corresponding dividend by multiplication, then, when

the divisor is filled, the division yields an integer. If, moreover, one multiplies something by the

numerator, the denominator must therefore divide in return [baochu]. To divide in return is to “divide

the dividend by the divisor”. Now, “the numerators are multiplied one by the other”, hence both

1 An independent English translation is also available (Crossley, Lun & Shen, 1999). As far as this passage is concerned,

the two translations differ significantly; our interpretation is based on Chemla’s translation and interpretation.

2 I edited out two similar examples (with proper fractions), as well as Li Chunfeng’s commentary.

denominators have to divide in return. Whence the multiplication of the two denominators, and the

division at one go [by their product].

The mere length and width of a rectangular field leave no room for a general explanation. Let someone

ask: “20 horses are valued at 12 jin of gold. Now 20 horses are sold and the proceeds are shared by

35 persons. How much does each one get?” The answer is 12/35 jin. To solve this by the procedure

for dividing into parts, take 12 jin of gold as dividend, and 35 persons as divisor. Now, change [the

problem] to: “5 horses are valued at 3 jin of gold; now 4 horses are sold, and [the proceeds] are shared

by 7 persons. How much does each one get?” Answer: each one gets 12/35 of jin. To solve this, you

need to homogenize these quantities (shu) of people and of gold; it is then completely similar to the

first problem, that of dividing into parts. If so, “multiply the numerators to make up the dividend”

does homogenize this quantity of gold; “multiply the numerators to make up the dividend” does

homogenize this quantity of people. Equalizing the denominators yields 20, but that plays no part: we

just need the homogenized. Moreover, when 5 horses are valued at 3 jin of gold, these are the lü in

whole numbers. If we express them with parts, 1 horse is valued at 3/5 jin of gold. Let 7 people sell

4 horses, 1 person does sell 4/7 of a horse. (…) As far as expression is concerned, that’s different; yet

as far as quantities are concerned, the three procedures boil down to the same.

Figure 1: (Chemla & Shuchun, 2004, 169-171). Free translation: R. Chorlay

Even if the procedure in quite easy to understand – and familiar to the reader – Liu Hui’s commentary

is probably hard to decipher. Even without a clear understanding of the details, one can probably see

at least two things: the goal of Liu is to justify the general procedure of the classic; he gives two

different justifications, one which does not rely on a semantic context nor on specific numbers, and

one which does.

Let us first focus on the first justification. Liu first distinguishes between two cases: a basic case in

which the fractions represent whole numbers; in this case, the procedure for multiplication is already

known. Of course, the goal is here to justify the new procedure – in the case where the divisions have

non-zero remainders – on the basis of the basic procedure. The word “procedure” is important, since

what is to be justified is the validity of an algorithm. More precisely, two sequences of calculations

are to be compared:

Sequence 1: divide a by b; divide c by d; multiply the two quotients.

Sequence 2: multiply a by c; multiply b by d; divide the first product by the second product.

If the two sequences relied on operations defined previously and independently, the goal would have

been to prove the universal equivalence of these two algorithms (that is, same entries yield the same

output). However, if one is not to beg the question, the first sequence actually contains the undefined

operation multiply, since the final multiplication may involve a multiplication of fractions3. So the

goal here is rather to justify that sequence 2 (which involves only well defined operations on integers)

3 When the remainder in the Euclidean division is non-zero, the quotient, the remainder and the dividend were read as a

mixed number. For instance, the division of 7 by 3 has quotient 2 and remainder 1, so its result is 2 and 1/3.

should be universally equivalent to sequence 1, thereby implicitly defining “multiply” in terms of

“multiply.”

To perform this, Liu uses terms which are specific to the commentary: whereas the Classic uses

“divide” and “multiply”, Liu uses “expand by multiplication”, and “divide in return” (baochu); these

terms point to the role of these operations. However, it is important to know that throughout the Nine

Chapters, “baochu” denotes a division whose role is to compensate for a multiplication by the same

number, so that these two steps of the procedure do not affect the final output of the sequence of

operations. On this basis, one can see that Liu’s justification involves a third sequence of operations:

Sequence 3: 1. multiply a by b, so that the first factor becomes ab / b, which is equal to a

2. multiply c by d, so that the second factor becomes cd / d, which is equal to c

3. multiply the two factors, which yields ac

4. divide by bd, which is the product of the two denominators

This sequence is not to be actually performed, it is mentioned for justificatory purposes only. Since

step 4 is a baochu-division, it compensates steps 1 and 2. The line of argumentation can thus be

reconstructed as: sequence 1 (which is yet undefined) should be universally equivalent to sequence

3, and sequence 3 boils down to sequence 2.

This line of argumentation may seem far-fetched. However, it is quite close to the following

contemporary proof of the following theorem: the only map f from ℚ × ℚ∗ which is ℤ-bilinear and

which is a prolongation of integer multiplication (i.e. if m and n are integers, then 𝑓(𝑚, 𝑛) = 𝑚 × 𝑛),

is 𝑓 (𝑎

𝑏,

𝑐

𝑑) =

𝑎×𝑐

𝑏×𝑑 (where a, b, c, d denote integers, with bd 0). The proof goes as follows:

(𝑏 × 𝑑) × 𝑓 (𝑎

𝑏,

𝑐

𝑑) = 𝑓 (𝑏 ×

𝑎

𝑏, 𝑑 ×

𝑐

𝑑) by ℤ-bilinearity

= 𝑓(𝑎, 𝑐) by a Lemma4 which relies only on addition in ℚ

= (𝑎 × 𝑐) (prolongation requirement)

Diving both sides by (bd) completes the proof. One could argue that, although Liu Hui states the

prolongation requirement quite explicitly, never does he say anything resembling the bilinearity

requirement. Even if this requirement is not explicit, it should be stressed that the bilinearity of the

operation being defined (i.e. multiply) can be rephrased as follows: multiplying either of the two

factors by and integer n should multiply the product by n; which is exactly the property which justifies

the role of the baochu division.

The second part of Liu Hui’s commentary clearly follows another line of argumentation. The first

part does not rely on a semantic context of interpretation, nor does it use specific numbers. The second

part, however, depends on a semantic context which Liu Hui introduces out of the blue after

dismissing the context provided by the Classic (rectangular areas), and uses the numbers from

Problem 19.

In the second part, several in-context problems are mentioned, and their relationships discussed.

4 𝑏 ×𝑎

𝑏=

𝑎

𝑏+ ⋯ +

𝑎

𝑏 (with 𝑏 terms) =

𝑏𝑎

𝑏= 𝑎 (assuming b ∈ ℕ∗).

Pb. 1: 20 horses are valued at 12 jin of gold [for all 20 horses]. Now 20 horses are sold and the

proceeds are shared by 35 persons. How much does each one get?

Pb. 2: 5 horses are valued at 3 jin of gold [for all 5]. Now 4 horses are sold and [the proceeds] are

shared by 7 persons. How much does each one get?

Problems 1 and 2 can be connected by a series of problems. Starting from Problem 2, on can consider:

Pb. 2’: 5x4 horses are valued at 3x4 jin of gold. Now 4 horses are sold and [the proceeds] are shared

by 7 persons. How much does each one get?

Although Pb2 and 2’ differ from a semantic viewpoint, they are equivalent in the following sense:

the numerical answers to Pb 2 and 2’ are equal, since the second parts are the same, and the answers

depends only on the ratio between horses and jin of gold. The argument would hold for any factor, it

so happens that “4” is more relevant than others for what follows5. This is quite explicit in the text,

since throughout the Nine Chapters the technical term lü denotes either a ratio, or numbers which are

to be considered up to multiplication by a common factor. The relationship between Pb2 and Pb 2’ is

the same as that between Pb2’ and Pb2’’:

Pb. 2’’: 5x4 horses are valued at 3x4 jin of gold. Now 4x5 horses are sold and [the proceeds] are

shared by 7x5 persons. How much does each one get?

Now, Pb 2’’ is the same as Pb 1, so their numerical answers are the same: 12/35.

But there is another way to solve Pb 2. The numerical answer to Pb 2 is the same as that of

Pb 2’’’: 1 horse is valued at 3/5 jin of gold. Now 4/7 horses are sold and [the proceeds] are shared

by 1 person. How much does this person one get?

We want to define multiplication of fractions so that the following property of integer multiplication

remains valid: total value = unit value times quantity. As a consequence, the product 3

5×

4

7 should be

the numerical answer to Pb 2’’’, hence to Pb 2 just as well, hence equal to 12/35.

A more formal summary would go as follows (we use ⊗ to denote to multiplication of fractions, to

be defined here in terms of ):

3

5⊗

4

7=

3 × 4

5 × 4⊗

4 × 5

7 × 5=

3 × 4

5 × 4⊗

4 × 5

7 × 5=

3 × 4

7 × 5

As for the extra-steps mentioned in the first argument, none of these intermediary operations are to

be performed when actually multiplying fractions; they are here for justificatory purposes only. This

formal summary does not do justice to the nature of the argumentation, since it is the context only

which justifies the equalities: dependence on ratios only (first equality); then reduction to a problem

involving only two integer data instead of four, and which can be solved by ordinary division. Just as

well, the fact that any given fraction can be freely replaced by any equivalent fraction can be justified

5 We will not comment on the use of technical terms such as “equalizing” and “homogenizing”, which Liu Hui introduced

in the context of the addition of fractions: 𝑎

𝑏+

𝑐

𝑑=

(𝑎𝑑)+ (𝑏𝑐) ← this is what you get by "homogenizing"

(𝑏𝑑) ← this is what you get by "equalizing" .

by the context; for instance, the unit price depends only on the ratio between the number of horses

and their total value.

Tools for the analysis of arguments in today’s textbooks

Reflecting on the Nine Chapters from a teacher-training perspective

The obscurity of the Chinese text probably makes it impossible to use in the classroom. In this paper,

we would like to show how the mathematical analysis of this text that we carried out in the first part

can help raise the awareness of prospective teachers as to several general features of argumentation

in rather informal contexts (this is our Hypothesis #1); and even supply them with tools which are

useful in everyday teaching, when they are to decide how to (somehow) justify some mathematical

definition or property (this is our Hypothesis #2)6.

The analysis of the Nine Chapters drew our attention to several facts, some of which bear specifically

on fraction multiplication, and some of which are of a more general scope:

There are several ways to justify the rule for the multiplication of fractions.

The use of letters is not the only way to express a general line of argumentation, as Liu Hui’s

first argument illustrates.

The rule for fraction multiplication is not only an equality between formulae; it can also be

seen as the equivalence of two different algorithms (one with two divisions followed by a

multiplication, and one with two multiplications followed by a division). This equivalence of

algorithm can be established using general arguments within the algorithmic framework (i.e.

arguments about algorithms), as opposed to the rewriting of formulae within an algebraic

framework.

Argumentation is not limited to properties or theorems: the choice of a definition (here: for

the multiplication fractions) can also be justified. With fraction multiplication we are dealing

with a case domain extension: some notion was already defined for a given class of objects;

the to-be-defined notion has to apply to a class of objects that encompasses the first class, and

to coincide with the former notion on the first class. In our case, a very weak form of

justification would be: this definition of fraction multiplication boils down to ordinary

multiplication (of integers) should the fractions denote integers (this could be called the

control case). The Nine Chapters suggests a stronger kind of justification: the extended

definition of “multiply” should preserve some structural properties of the former definition,

properties that we value. For instance: bilinearity; or the validity of the formula “total value =

unit value times quantity”. On the other hand, we are ready to give up on some properties,

such as: the product of two numbers is greater than or equal to either of them. To some extent,

when justify the choice of a definition, these required properties play the part that hypotheses

play in the proof of a theorem.

In some cases, from a mathematical viewpoint, these requirements completely characterize

the new notion. In these cases, whether or not this uniqueness property should be proved in

6 The decisions as to how to justify a given definition or property is closely related to the decision as to whether or not it

should be, and to the knowledge of ways to justify it (can it be defined, or not, in a given mathematical context?). We

will not have time to discuss the interplay of these three aspects in this short paper.

the classroom calls for another decision. Even if the teacher thinks the uniqueness aspect

cannot / need not be made explicit, the Nine Chapters suggest another lead. Making explicit

the properties that we want to extend also has a heuristic value, suggesting the path to a

definition of the extended notion.

Liu Hui’s second line of argumentation also suggests many fruitful leads, which we can only

mention in passing for lack of space:

o A rather unusual form of argument: proving that two formulae or two algorithms are

universally equal / equivalent by exhibiting a class of in-context problems for which

the answer is unique – for any given set of initial data –, and for which – for reasons

that can be spelled out – both formulae or algorithms work out the solution. Of course,

this form of argumentation – that can be called semantic immersion – raises questions

as to the context-dependence of the argument. For instance, multiplication plays a part

in the context of commercial transactions and in the context of area calculation, and

nothing proves that the fact that the relevant operations coincide in both contexts when

the data are integers will still hold with all rational data.

o Liu Hui’s second argument is not only in-context (a horse-deal), it is also an arguments

which clearly claims full generality while dwelling on a single specific case, that of

3/5 times 4/7. This raises the well-kown but deep questions pertaining to the notion of

generic example (Balachef, 1987, 157). Let us mention two such questions: (1) from

a mathematical viewpoint, which conditions guarantee the minimal level of non-

specificity that an example should enjoy to be potentially seen as generic (for instance:

dealing with 1/5, 7/5, or 15/5 instead of 3/5 would probably make this case less

generic)? (2) The genericity of a case is not a mathematical property but is a

multilayered property of the relationship between it and the students; it involves

pragmatic aspects (students should be able to adapt the reasoning to other cases), and

epistemological aspects (students should regard this case as a mere representative of

all cases, thus providing some form a general argument); what are the general

conditions for a didactical genesis of this relationship?

Field-work

We shall end with some applied work. The following three documents are extracts from contemporary

French textbooks or standard teacher-training documents, for students in the second year of middle-

school (Figures 2 and 3) or in the last year of primary school (Figure 4). In cases 2 and 3 they are

meant for students who are somewhat familiar with fractions and their addition, but with no

knowledge of multiplication. As to Figure 4, students are familiar with integer multiplication, and

with multiplication and division of decimals by powers of 10. These extracts are what most French

textbooks call “introductory activities”, in which students are to experience a new notion from a

hands-on, guided but non-dogmatic approach. We suggest these documents be analyzed in TWG12

in order to test Hypotheses #1 and #2.

Figure 2: Two ways to work out a rectangular area (Malaval, 2014, 69)

Figure 3: A justification in terms of combination of operators, and a conjecture relying on the

control-case of decimal fractions (Mante, 2006, 50)

Figure 4: Generalizing multiplication from integers to decimals using bilinearity7

Conclusion

The goal of this short paper was to contribute to the ongoing reflection on the use of original historical

sources in the classroom or in teacher-training; a reflection which bears both on the goals of this use,

and on the relevant ways to use such documents.

Within this general field of research, the main features of this specific contribution are: we dealt with

teacher-training only, since we do not think this excerpt from the Nine Chapters can be used with

7 Document retrieved from a teacher-training slideshow: http://docslide.fr/documents/techniques-operatoires-cycles-2-et-

3-multiplication-jean-luc-despretz-cpc-landivisiau-avril-2010-lacquisition-des-mecanismes-en-mathematiques-est-

toujours.html (slide 22). Accessed September 14, 2016.

students who learn to multiply fractions; we endeavoured to show that even for such documents, they

can be used in teacher-training for other purposes than simply teaching some history of mathematics,

or enriching teacher’s image of mathematics. As teachers, it is or will be part of their everyday work

to analyze/assess teaching documents and make decisions on the basis of this analysis/assessment.

Among these documents, some will be of an argumentative nature: sometimes in the form of pretty

formal and academic proofs; more often than not, these documents will explicitly display or rather

implicitly point to other forms of argumentation. We think the conceptual analysis – an analysis which

has a mathematical component and an epistemological component, and which is to be distinguished

from both the historical analysis and the didactical analysis – of some historical documents can give

teacher-trainers opportunities to make explicit some tools which can be of constant use in the analysis

and assessment of argumentative documents.

In this paper, we dealt with pretty informal arguments and elementary mathematics. However, we do

not think this general scheme is specific to such contexts, since this work is a continuation of our

CERME8 paper (Chorlay, 2013), in which a similar approach was used in a formal and rather

advanced mathematical context (proof of the relationship between the variations of a function and the

sign of its derivative).

References

Balacheff, N. (1987). Processus de preuve et situations de validation. Educational Studies in

Mathematics, 18, 147−176.

Chemla, K., Shuchun, G. (2004). Les neuf chapitres : Le classique mathématique de la Chine

ancienne et ses commentaires. Paris: Dunod.

Chorlay, R. (2013). The making of a proof-chain: Epistemological and didactical perspectives. In B.

Ubuz, Ç. Haser, & M.-A. Mariotti (Eds.), Proceedings of the Eighth Congress of the European

Society for Research in Mathematics Education (pp. 106−115). Ankara, Turkey: Middle East

Technical University and ERME.

Crossley, N., Lun, A., Shen, K. (1999). The nine chapters on the mathematical art: Companion and

commentary. Oxford: O.U.P.

Malaval, J. (Ed.) (2014). Transmath 5ème. Paris: Nathan.

Mante, M. (Ed.) (2006). Triangle 5ème. Paris: Hatier.